In connection with driving rules in a CMG (Control Moment Gyro) system having redundancy, there are a large number of reports or technical papers directed to the ways of avoiding or getting rid of a singularity. Qualitatively, a driving rule for maintaining the input output gain always high is described in Yoshikawa, T., “A Steering Law for Three Double Gimbals Control Moment Gyro System”, NASA TM-X-64926 (1975) (Non-patent Document 1).
As described in the Non-patent Document 1, the “Gradient Method” (referred to as the GM method below) is a gimbal driving rule for making a judgment of positive time change dV/dt in volume of the input output gain from the instantaneous gimbal angle information with V as the volume cost function in the input output gain. This enables to avoid the off-line planning of complicated gimbal angle, thereby easily and effectively achieving to avoid singularity. Because of these reasons, the GM is the most popular driving rule for an attitude control of space craft bodies such as artificial satellites and the like and is the basis of studies on singularity avoidance.
It is to be noted that there are two kinds of singularity; one is an “passable singularity” that is capable of avoiding or getting rid of the condition by a null motion operation, while the other is a “impassable singularity” that is unable to avoid or get rid of the condition by such null motion operation. In case of a pyramid configuration system comprising four SG-CMGs (Single Gimbal-CMGs), there is a “impassable singularity” and the GM method is impossible to ensure its operation at the “impassable singularity”. Although it is possible to get rid of the “passable singularity” by the null motion operation, the output torque becomes zero in this case.
Consequently, various theoretical studies have been made in order to cope with the “impassable singularity”. For example, proposed are an SR-inverse method as disclosed in Nakamura, Y. and Hanafusa H., “Inverse Kinematic Solutions with Singularity Robustness for Robot Manipulator Control”, Journal of Dynamic Systems, Measurement and Control, Vol. 108, September, (1986), pp. 163-171 (Non-patent Document 2), a generalized SR-inverse method as disclosed in Bong Wie, David Bailey and Christopher Heiberg, “Singularity Robust Steering Logic for Redundant Single-Gimbal Control Moment Gyros”, AIAA Journal of Guidance, Control and Dynamics, Vol. 24, No. 5, (2001) pp. 865-872 (Non-patent Document 3) and the like. All of such prior art are methods for avoiding or getting rid of the singularity by causing small torque disturbance at or near the “impassable singularity”. However, there is a need for performance tradeoff due to torque disturbance.
On the other hand, Kurokawa H., “Constrained Steering Law of Pyramid-Type Control Moment Gyros and Ground Test”, AIAA Journal of Guidance, Control and Dynamics, Vol. 20, No. 3, (1997), pp 445-449 (Non-patent Document 4) proposes a driving rule that provides a constraint condition to achieve an operation not to reach the “impassable singularity”. However, it is a countermeasure at the immediate location of the “impassable singularity” but not directed to avoid undesired situations at the areas both at and near the singularity including the “passable singularity” where the input output gain is insufficient. As a result, if this particular driving rule is employed, it is impossible to deny the existence of incapability to output sufficient torque. Moreover, when it reaches the “passable singularity”, it is possible that the output torque becomes zero temporarily in a certain direction.
Accordingly, the aforementioned driving rules can be applied only to cases where the necessary torque is very small such as maintaining an attitude other than at and near the “passable singularity”. However, in case of high speed attitude change that continuously requires a large feed forward torque, countermeasures near the singularity are insufficient in the input output gain for achieving torque necessary for attitude change.
It is useful to previously perform off-line path planning of the gimbal angle operation in order to avoid singularity and secure sufficient input output gain. However, an off-line path planning as disclosed, for example, in Paradiso J., “Application of a Directed Search to Global Steering of Single Gimballed CMGs”, CSDL-P-3014, Proc. Of the AIAA Guidance, Navigation and Control Conference, New Orleans, La., August, (1991) 12-15, AIAA Paper 91-2718 (Non-patent Document 5) is very complicated and causes a calculation load problem.